Right, I follow... So my question could really be re-phrased as: when t is complex we have a bona fides LT, but what is it that we are doing when in your first integral t is real? Are we just "dotting" the function with an exponential to see how it decays?
The Laplace transform gives information about the exponential components in a function, as well as oscillatory components. To do so there is a need for the complex plane (complex exponentials).
I get why the MGF of a distribution is very useful (moment extraction and classification of the...